My paper “Simple groups stabilizing polynomials“, written with Bob Guralnick, has appeared in Forum of Mathematics Pi.  In it, we show that, if a simple algebraic group G stabilizes a polynomial function f on a vector space V, then with a very short list of exceptions G is actually the identity component of the stabilizer of f.  We give various concrete applications of this general fact.  The proof relies on knowing that there are very few groups H containing G, and our core result, which shows that for only a very short list of those do the quotient spaces V/G and V/H have the same dimension.